5.9 Double Trouble with Multiple Roots

   

 
Java Number Cruncher: The Java Programmer's Guide to Numerical Computing
By Ronald  Mak

Table of Contents
Chapter  5.   Finding Roots


If the graph of function f ( x ) touches or crosses the x axis tangentially, then there are multiple roots at that point. In other words, if f ( x r ) = 0 and f '( x r ) = 0, then x r is a multiple root. For example, 2 is a double root of the function

graphics/05equ17.gif


Multiple roots are problematic for Newton's algorithm and the secant algorithm because the derivative value is 0. The function plot does not cross the x axis at an even degree (double, quadruple, etc.) multiple root. Figure 5-4 shows that the function f ( x ) = x 2 - 4 x + 4 is always positive, so the bracketing algorithms, which require a sign change, won't work, either.

Figure 5-4. The graph of f ( x ) = x 2 - 4 x + 4 with a double root at x = 2.

graphics/05fig04.jpg


   
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Java Number Cruncher. The Java Programmer's Guide to Numerical Computing
Java Number Cruncher: The Java Programmers Guide to Numerical Computing
ISBN: 0130460419
EAN: 2147483647
Year: 2001
Pages: 141
Authors: Ronald Mak

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